Kernels with complete Nevanlinna-Pick factors and the characteristic function

Tirthankar Bhattacharyya, Abhay Jindal

Published 2023-07-20Version 1

The Sz.-Nagy Foias charateristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna- Pick (CNP) kernels. Indeed, an irreducible unitarily invariant kernel $k$ on the $d$-dimensional Euclidean unit ball admits a characteristic function if and only if $k$ is a CNP kernel. We are intrigued by recent constructions of the characteristic function for kernels which are not CNP. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector valued Drury-Arveson space (thus the Hardy space when $d = 1$). We show that the construction of a characteristic function is always possible when $k$ has a CNP factor $s$.